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Turbulent Kinetic Energy (TKE). An equation to describe TKE is obtained by multiplying the momentum equation for turbulent flow times the flow itself (scalar product). Total flow = Mean plus turbulent parts =. Same for a scalar:.
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Turbulent Kinetic Energy (TKE) An equation to describe TKE is obtained by multiplying the momentum equation for turbulent flow times the flow itself (scalar product) Total flow = Mean plus turbulent parts = Same for a scalar:
- Use these properties of turbulent flows in the Navier Stokes equations • The only terms that have products of fluctuations are the advection terms • All other terms remain the same, e.g.,
0 are the Reynolds stresses arise from advective (non-linear or inertial) terms
Multiplying turbulent flow times ui and dropping the primes fluctuating strain rate Turbulent Kinetic Energy (TKE) Equation Total changes of TKE Transport of TKE Shear Production Buoyancy Production Viscous Dissipation Transport of TKE. Has a flux divergence form and represents spatial transport of TKE. The first two terms are transport of turbulence by turbulence itself: pressure fluctuations (waves) and turbulent transport by eddies; the third term is viscous transport
interaction of Reynolds stresses with mean shear; represents gain of TKE represents gain or loss of TKE, depending on covariance of density and w fluctuations represents loss of TKE
In many ocean applications, the TKE balance is approximated as:
Vertical Shears (vertical gradients) Shear production from bottom stress z u bottom
z W Vertical Shears (vertical gradients) u Shear production from wind stress
Vertical Shears (vertical gradients) Shear production from internal stresses z u1 u2 Flux of momentum from regions of fast flow to regions of slow flow
Near the bottom Bottom stress: Parameterizations and representations of Shear Production
(Monismith’s Lectures) Law of the wall may be widely applicable
(Monismith’s Lectures) Ralph Obtained from velocity profiles and best fitting them to the values of z0 and u*
With ADCP: and θ is the angle of ADCP’s transducers -- 20º Lohrmann et al. (1990, J. Oc. Atmos. Tech., 7, 19) Shear Production from Reynolds’ stresses Mixing of property S Mixing of momentum
Souza et al. (2004, Geophys. Res. Lett., 31, L20309) Day of the year (2002)
S2 > S1 S1, T1 T2 > T1 S2, T2 Buoyancy Production from Cooling and Double Diffusion
Layering Experiment http://www.phys.ocean.dal.ca/programs/doubdiff/labdemos.html
Data from the Arctic From Kelley et al. (2002, The Diffusive Regime of Double-Diffusive Convection)
Dissipation from strain in the flow (m2/s3) (Jennifer MacKinnon’s webpage)
Production of TKE From: Rippeth et al. (2003, JPO, 1889) Dissipation of TKE
Other ways to determine dissipation (indirectly) Az (Monismith’s Lectures)
(responsible for dissipation of TKE) Inertial subrange – transfer of energy by inertial forces (Monismith’s Lectures)
P equilibrium range inertial dissipating range Kolmogorov’s K-5/3 law (Monismith’s Lectures)
Stratification kills turbulence In stratified flow, buoyancy tends to: i) inhibit range of scales in the subinertial range ii) “kill” the turbulence